On Picard Groups of Perfectoid Covers of Toric Varieties

TL;DR

This paper investigates the Picard group of perfectoid covers of toric varieties, showing an isomorphism with the original Picard group after inverting p, and computes line bundle cohomology with vanishing results.

Contribution

It establishes a canonical isomorphism between the Picard group of the perfectoid cover and the original Picard group after inverting p, extending previous results to a broader class of toric varieties.

Findings

Picard group of perfectoid cover is isomorphic to the original Picard group after inverting p

Computed cohomology of line bundles on the perfectoid cover

Established analogs of Demazure and Batyrev-Borisov vanishing theorems

Abstract

Let $X$ be a proper smooth toric variety over a perfectoid field of prime residue characteristic $p$. We study the perfectoid space $\mathcal{X}^{perf}$ which covers $X$ constructed by Scholze, showing that $\text{Pic}(\mathcal{X}^{perf})$ is canonically isomorphic to $\text{Pic}(X)[p^{-1}]$. We also compute the cohomology of line bundles on $\mathcal{X}^{perf}$ and establish analogs of Demazure and Batyrev-Borisov vanishing. This generalizes the first author's analogous results for "projectivoid space".